Understanding Normal Distribution: Calculating Scores Between 60 and 75 on a Test

When dealing with test scores or any set of data, understanding the normal distribution can be very useful. This article demonstrates how to calculate the number of test takers who scored between 60 and 75, given a mean score, standard deviation, and total number of test takers. We will go through each step and explain the mathematical concepts involved.

Given Data

Mean Score (μ): 70 Standard Deviation (σ): 15 Total Test Takers: 1000

Step-by-Step Calculation

Step 1: Calculate Z-scores

The Z-score formula is:

Z frac{{X - mu}}{{sigma}}

Where:

X is the score mu is the mean sigma is the standard deviation

For 60:

Z_{60} frac{{60 - 70}}{{15}} frac{{-10}}{{15}} -frac{2}{3} approx -0.67

For 75:

Z_{75} frac{{75 - 70}}{{15}} frac{{5}}{{15}} frac{1}{3} approx 0.33

Step 2: Find the Proportion of Scores

Next, we need to find the cumulative probabilities for these Z-scores using the standard normal distribution table or a calculator.

Cumulative probability for Z_{60} approx -0.67: This is approximately 0.2514.

Cumulative probability for Z_{75} approx 0.33: This is approximately 0.6293.

Step 3: Calculate the Proportion Between 60 and 75

To find the proportion of test takers scoring between 60 and 75, we subtract the cumulative probability of Z_{60} from that of Z_{75}:

P60 le X le 75 P(Z le 0.33) - P(Z le -0.67) approx 0.6293 - 0.2514 0.3779

Step 4: Calculate the Number of Test Takers

Now, we multiply the proportion by the total number of test takers (1000):

Number of test takers 0.3779 times 1000 approx 377.9

Since the number of people must be a whole number, we round to:

Number of test takers approx 378

Conclusion

Therefore, approximately 378 test takers scored between 60 and 75.

Caution: Validity of the Assumptions

It is important to note that if a normal distribution is assumed, the exact calculation can be done using a standard normal distribution table. However, the given data contradicts this assumption. If you create a dataset of 1000 entries with a mean of 70 and a standard deviation of 15, with 500 scores of 55 and 500 scores of 85, the number of test takers scoring between 60 and 75 is zero. This demonstrates that the normal distribution may not accurately represent the true distribution of the data.